Simplifying Linear Algebra: Techniques, Solutions, and Examples

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row echelon form. Solve the system. Assume that the variables are named $x_1,x_2,...x,...$ ,… from left to right. 

Is there a relationship between Facebook use and age among college students? The following two-way table displays data for the 219 students who responded to the survey. $\begin{array}{cc}& \text{Age}\\ \text{Facebook user?}& \begin{array}{lccc}& \begin{array}{c}\text{ Younger }\\ (18-22)\end{array}& \begin{array}{c}\text{ Middle }\\ (23-27)\end{array}& \begin{array}{c}\text{ Older }\\ (28\text{ and up)}\end{array}\\ \text{ Yes }& 78& 49& 21\\ \text{ No }& 4& 21& 46\end{array}\end{array}$

What percent of the students who responded were older Facebook users?

The reduced row echelon form of a system of linear equations is given.Write the system of equations corresponding to the given matrix. Use x, y; or x, y, z; or $x_1,x_2,x_3,x_4$ as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. $\left[\begin{array}{ccccc}1& 0& 0& 4& 2\\ 0& 1& 1& 3& 3\\ 0& 0& 0& 0& 0\end{array}\right]$

The reduced row echelon form of a system of linear equations is given.Write the system of equations corresponding to the given matrix. Use x, y; or x, y, z; or $x_1,x_2,x_3,x_4$ as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. $\left[\begin{array}{cccc}1& 0& 4& 4\\ 0& 1& 3& 2\\ 0& 0& 0& 0\end{array}\right]$

The row echelon form of a system of linear equations is given.
(a)Write the system of equations corresponding to the given matrix.Usex,y;orx,y,z;or $x_1,x_2,x_3,x_4$ asvariables.
(b)Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. $\left[\begin{array}{ccc}1& 2& 5\\ 0& 1& -1\end{array}\right]$

Write the vector form of the general solution of the given system of linear equations.
$x_1+2x_2-x_3+x_4=0$

$x+y+2z=3$
$3x+2y+2z=4$
$x+y+3z=5$

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row echelon form. Solve the system. Assume that the variables are named $x_1,x_2,...$ form left to right. $\left[\begin{array}{ccccc}1& 0& 5& 3& 2\\ 0& 1& -2& 4& -7\\ 0& 0& 1& 1& 3\end{array}\right]$

The reduced row echelon form of the augmented matrix of a system of three linear equations in three variables must be of the form

$\left[\begin{array}{cccc}1& 0& 0& a\\ 0& 1& 0& b\\ 0& 0& 1& c\end{array}\right]$

where a, b, c, are real numbers. Justify your answer.

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Tell whether the system has one solution, no solution, or infinitely many solutions. Write the solutions or, if there is no solution, say the system is inconsistent.

$\left[\begin{array}{ccccc}1& 0& 0& |& -1\\ 0& 1& 0& |& 3\\ 0& 0& 1& |& 4\\ 0& 0& 0& |& 0\end{array}\right]$

find an equation of the form $y=a{x}^2+bx+c$, whose graph passess through the points (2, 3), (-2, 7) and (1, -2). You must use a system of linear equations in three variables for this problem.

The coefficient matrix for a system of linear differential equations of the form y′=Ay has the given eigenvalues and eigenspace bases. Find the general solution for the system.
${\displaystyle {\lambda }_1=1\Rightarrow \left\{\left[\begin{array}{c}2\\ -1\end{array}\right]\right\},{\lambda }_2=3\Rightarrow \left\{\left[\begin{array}{c}3\\ 1\end{array}\right]\right\}}$

Solve the system of linear equations using the substitution method.
$x+y-z=4$

$3x+2y+4z=17$

$-x+5y+z=8$